Objectives (5 - 7 minutes)

1. Develop students' understanding of the concept of probability in successive events, explaining how to calculate the probability of different events occurring in sequence.

2. Help students understand how to use tree diagrams to visualize and calculate the probability of successive events.

3. Foster students' ability to apply the concept of probability to real-world situations, encouraging them to think critically and solve complex problems.

Secondary Objectives:

• Stimulate active participation of students in the class, whether through questions and answers, group discussions, or practical activities.

• Promote an interactive and collaborative learning environment where students can freely share their ideas and doubts.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher should start the lesson by briefly reviewing the probability concepts already learned, such as coin and dice tossing. They should remind students about the concept of an event (an outcome or set of outcomes) and sample space (the set of all possible outcomes). This review is crucial for students to become familiar with the necessary terminology for the lesson topic.

2. Problem Situations: The teacher should then present two problem situations involving the probability of successive events:

   • Situation 1: 'If you have two coins and toss them at the same time, what is the probability of getting heads on the first coin and tails on the second?'

   • Situation 2: 'If you have a bag with 3 balls (2 red and 1 blue) and draw a ball without looking, what is the probability of getting a red ball? And if, then, without returning the first ball to the bag, you draw another ball, what is the probability of getting a blue ball?'

3. Contextualization: The teacher should explain the importance of probability in successive events, highlighting that this concept is applied in various areas such as statistics, gambling games, weather forecasting, among others. The teacher can also mention examples of using probability in everyday situations, such as in a card game, predicting sports results, in medicine (calculating risks of genetic diseases, for example), among others.

4. Introduction to the Topic: To spark students' interest, the teacher can present two curiosities related to the theme:

   • Curiosity 1: 'Did you know that the probability of winning the Mega Sena, by hitting all 6 numbers, is approximately 1 in 50 million? This means that if you play once a week, it would take an average of 960,000 years to win!'

   • Curiosity 2: 'And what if I told you that when tossing a fair coin 10 times, the probability of getting 10 heads is the same as getting 5 heads and 5 tails? This happens because each toss is an independent event, meaning the result of one toss does not affect the result of the next.'

The teacher should conclude the Introduction by presenting the lesson objective and promising that by the end of the lesson, students will be able to calculate the probability of successive events and apply this concept to different situations.

Development (20 - 25 minutes)

1. Theory: Concept of Successive Events (5 - 7 minutes)

   • The teacher should start by explaining that successive events are those that occur one after the other.

   • It should be emphasized that, unlike independent events (like coin or dice tossing), in successive events the result of one event can affect the results of the following events.

   • To illustrate this point, the teacher can use the example of tossing a coin and drawing a ball from a bag, which were presented in the Introduction.

   • It is important to emphasize that in successive events, the probability of an event occurring depends on the results of previous events.

2. Theory: Tree Diagrams (5 - 7 minutes)

   • The teacher should introduce the concept of tree diagrams as a useful tool to visualize and calculate the probability of successive events.

   • They should explain that a tree diagram is a graphical representation of a sequence of events, where each branch represents an event and the probability of that event occurring is indicated next to the branch.

   • The teacher should show how to build a tree diagram for the problem situations presented in the Introduction. They should explain that the probability of an event occurring is the product of the probabilities of the branches leading to that event.

   • To facilitate understanding, the teacher can build a tree diagram on the board and calculate the probability of an event occurring from that diagram.

3. Practice: Probability Calculation (5 - 7 minutes)

   • The teacher should propose that students, in groups, calculate the probability of different successive events occurring using tree diagrams.

   • The teacher should provide students with different problem situations (for example, the probability of drawing different cards from a deck, the probability of getting different results in a sequence of coin tosses, etc.) and guide them in constructing the tree diagrams and calculating the probabilities.

   • The teacher should move around the classroom, assisting groups that encounter difficulties and clarifying doubts.

4. Theory: Independent and Dependent Events (5 - 7 minutes)

   • The teacher should review the concept of independent events and introduce the concept of dependent events.

   • They should explain that in independent events, the probability of an event occurring is not affected by the results of previous events.

   • On the other hand, in dependent events, the probability of an event occurring depends on the results of previous events.

   • The teacher should show how to identify whether a sequence of events is independent or dependent.

   • They should also emphasize that in dependent events, the probability of an event occurring can be calculated by multiplying the probabilities of the individual events, while in independent events the probability of an event occurring is the same in each attempt.

Return (8 - 10 minutes)

1. Concept Review (2 - 3 minutes): The teacher should start the Return by reviewing the main concepts presented in the lesson. They should remind students about what successive events are and how to calculate the probability of these events occurring. The teacher should also recap the use of tree diagrams and the difference between independent and dependent events. This review is important to consolidate students' learning and prepare them for the next stage of the Return.

2. Connection to Practice (2 - 3 minutes): The teacher should then connect the theory presented with practice. They should review the problem situations solved in groups and explain how the theoretical concepts were applied to calculate the probabilities. The teacher should also highlight how understanding these concepts can help students solve real-world problems involving the probability of successive events, such as weather forecasting, gambling games, among others.

3. Individual Reflection (2 - 3 minutes): The teacher should propose that students reflect individually on what they learned in the lesson. They should ask questions like:

   1. 'What was the most important concept you learned today?'

   2. 'What questions have not been answered yet?'

   3. 'How can you apply what you learned today in everyday situations?'

   The teacher should encourage students to write down their answers and share their reflections with the class. This reflection stage is crucial for students to internalize what they learned and for the teacher to identify possible gaps in students' understanding.

4. Feedback and Clarification of Doubts (2 - 3 minutes): Finally, the teacher should open space for students to express their doubts and comments. They should answer students' questions, clarify any misunderstandings, and provide feedback on students' performance during the lesson. The teacher should encourage students to be honest in their evaluations and to express their opinions freely. This will help the teacher improve the planning of their future lessons and adapt their teaching to individual students' needs.

Conclusion (5 - 7 minutes)

1. Summary of Contents (1 - 2 minutes): The teacher should start the Conclusion by recalling the main points covered during the lesson. They should summarize the concept of successive events, the importance of tree diagrams for calculating the probability of these events, and the difference between dependent and independent events.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): Next, the teacher should emphasize how the lesson connected the theory, practice, and applications of the probability concept in successive events. They should reinforce that, in addition to learning the theory behind probability calculations, students also had the opportunity to apply this knowledge to real situations. The teacher should mention again the problem situations discussed during the lesson and explain how the theory helped solve these problems.

3. Extra Materials (1 - 2 minutes): The teacher should then suggest some extra materials for students who wish to deepen their understanding of the lesson topic. These materials may include math books, online learning websites, explanatory videos, among others. The teacher should encourage students to explore these materials on their own and to bring any doubts or questions that arise during independent study to future lessons.

4. Applications in Everyday Life (1 minute): To conclude, the teacher should highlight the relevance of the probability concept in successive events for students' daily lives. They should recall some of the applications discussed during the lesson, such as weather forecasting, gambling games, among others. The teacher should emphasize that understanding this concept can help students make more informed decisions in various situations, from choosing the best strategy in a game to understanding the implications of a medical decision.

5. Closure (1 minute): Finally, the teacher should end the lesson by thanking the students for their participation and reinforcing the importance of continuous study and individual effort for success in learning. They should encourage students to continue practicing what they learned and to bring any doubts or difficulties to future lessons.